# The value is=?

Jun 26, 2018

The Right Option is $\left(4\right) : \frac{5 \sqrt{6}}{12}$.

#### Explanation:

Let, $a , b \in {\mathbb{R}}^{+} \text{ such that, } a > b$.

Their $\text{AM"=(a+b)/2," and the GM} = \sqrt{a b}$.

It is given that, $\frac{a + b}{2} = 5 \sqrt{a b}$.

$\therefore \left(a + b\right) = 10 \sqrt{a b}$.

$\therefore {\left(a + b\right)}^{2} = 100 a b \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left({\ast}^{1}\right)$.

Further, ${\left(a - b\right)}^{2} = {\left(a + b\right)}^{2} - 4 a b$.

$= 100 a b - 4 a b \ldots \ldots \ldots . \left[\because , \left({\ast}^{1}\right)\right]$.

$\therefore {\left(a - b\right)}^{2} = 96 a b \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left({\ast}^{2}\right)$.

From $\left({\ast}^{1}\right) \mathmr{and} \left({\ast}^{2}\right)$, we have,

${\left(a + b\right)}^{2} / {\left(a - b\right)}^{2} = \frac{100 a b}{96 a b} = \frac{25}{24}$.

$\therefore \frac{a + b}{a - b} = + \sqrt{{5}^{2} / \left({2}^{2} \cdot 6\right)} \ldots \ldots \ldots \ldots \left[\because , \left(a - b\right) > 0\right]$,

$= \frac{5}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$.

$\Rightarrow \frac{a + b}{a - b} = \frac{5 \sqrt{6}}{12}$, showing that,

The Right Option is $\left(4\right) : \frac{5 \sqrt{6}}{12}$.